Page 276 - Numerical Methods for Chemical Engineering

A spherical catalyst pellet 265

eterna id c (R) = c AS
A
T(R) = T S
c (r)
A
cte T(r)
R

sid catast eet
A → B

Figure 6.4 Nonisothermal reaction within a catalyst pellet with internal diffusion and thermal con-
duction.

Chemical reaction and diffusion in a spherical catalyst pellet

Consider the case of a nonisothermal reaction A → B occurring in the interior of a spherical
catalyst pellet of radius R (Figure 6.4). We wish to compute the effect of internal heat and
masstransferresistanceuponthereactionrateandtheconcentrationandtemperatureprofiles
within the pellet. If D A is the effective binary diffusivity of A within the pellet, and we have
first-order kinetics, the concentration profile c A (r) is governed by the mole balance
d
2 dc A 2
r D A − r [k(T )c A ] = 0 (6.32)
dr dr
Similarly, if λ is the effective thermal conductivity of the pellet, the temperature profile
T (r) is governed by the enthalpy balance
d 2 dT 2
r λ + r (− H)[k(T )c A ] = 0 (6.33)
dr dr
Neglecting external heat or mass transfer resistance, we have known values of the concen-
tration and temperature at the surface, r = R. At the pellet center, we use the symmetry
conditions dc A /dr = dT/dr = 0. Thus, we solve (6.32) and (6.33) subject to the boundary
conditions

BC 1 c A (R) = c AS T (R) = T S (6.34)

dc A dT
BC 2 = 0 = 0 (6.35)
dr r=0 dr r=0
The temperature dependence of the rate constant is

E a T S
k(T ) = k(T S )exp − − 1 (6.36)
RT S T
This BVP introduces several new issues: (1) nonCartesian (spherical) coordinates, (2) more
than one coupled PDE, and (3) a BC at r = 0 that specifies the local value of the gradient (a
von Neumann-type boundary condition). Also, experience tells us that when internal mass
transfer resistance is strong, reaction only occurs within a thin layer near the surface over
which the local concentration of A drops rapidly to zero. Thus, we use a computational